Bounds on List Decoding of Rank-Metric Codes

Wachter-Zeh, Antonia

HAL, hal-00904783 / Harvested from HAL

Résumé

So far, there is no polynomial-time list decoding algorithm (beyond half the minimum distance) for Gabidulin codes. These codes can be seen as the rank-metric equivalent of Reed-Solomon codes. In this paper, we provide bounds on the list size of rank-metric codes in order to understand whether polynomial-time list decoding is possible or whether it works only with exponential time complexity. Three bounds on the list size are proven. The first one is a lower exponential bound for Gabidulin codes and shows that for these codes no polynomial-time list decoding beyond the Johnson radius exists. Second, an exponential upper bound is derived, which holds for any rank-metric code of length n and minimum rank distance d. The third bound proves that there exists a rank-metric code over F-qm of length n <= m such that the list size is exponential in the length for any radius greater than half the minimum rank distance. This implies that there cannot exist a polynomial upper bound depending only on n and d similar to the Johnson bound in Hamming metric. All three rank-metric bounds reveal significant differences to bounds for codes in Hamming metric.

Détail

Publié le : 2013-07-05
Classification:  Gabidulin codes,  list decoding,  Constant-rank codes,  rank-metric codes,  [INFO.INFO-IT]Computer Science [cs]/Information Theory [cs.IT],  [MATH.MATH-IT]Mathematics [math]/Information Theory [math.IT]